I have a partitioned matrix ($2n \times 2n$) like this. \begin{align} \mathbf{M}= \begin{pmatrix} \mathbf{X} & \mathbf{Y} \\ \mathbf{Y^{\dagger}} & -\mathbf{X} \end{pmatrix} \end{align}
The matrices $X$ and $Y$ are $n\times n$ dimensional matrices. $\mathbf M$ is a Hermitian matrix. Generally, it's very hard to solve the eigenvalue problem for $\mathbf M$ and get the exact expressions. If it were possible to block diagonalize the matrix like \begin{align} \mathbf M = \mathbf m \oplus (-\mathbf m) \end{align} Then one could just solve the eigenvalue problem of $\mathbf m$ and found all the eigenvalues of $\mathbf M$. For when $\mathbf X = \mathbf 0$ then the solution can be obtained using this method like here . In my case, I have a diagonal matrix as my $\mathbf X$. I tried looking up diagonalization of partitioned matrices but I had no luck.
Is there any way I can at least block diagonalize $\mathbf M$ ?